André was correct in claiming that phaneroscopy must involve a transition from experience in the phaneron to something that can be interpreted by ordinary common sense. But he made a mistake in claiming that it could not be mathematical.
See below for five quotations by CSP, which make the following points. 1. >From the first quotation, a diagram is an "icon, which exhibits a similarity or analogy to the subject of discourse." 2. "we construct an icon of our hypothetical state of things and proceed to observe it... We not only have to select the features of the diagram which it will be pertinent to pay attention to, but it is also of great importance to return again and again to certain features." 3. A diagram may be "a concrete, but possibly changing, mental image of such a thing as it represents." 4. "We form in the imagination some sort of diagrammatic, that is, iconic, representation of the facts, as skeletonized as possible." 5. "Diagrammatic reasoning is the only really fertile reasoning." For more quotations by Peirce, search CP for the 538 instances of 'diagram' with or without various suffixes. They show that diagrams are "iconic", they are fundamental for "necessary reasoning" in mathematics, and they can be understood by anybody that André calls "the rest of us". John ------------------------------ 1. there are three kinds of signs which are all indispensable in all reasoning; the first is the diagrammatic sign or icon, which exhibits a similarity or analogy to the subject of discourse. [second is index; third is symbol] (CP 1.369) 2. All necessary reasoning without exception is diagrammatic. That is, we construct an icon of our hypothetical state of things and proceed to observe it. This observation leads us to suspect that something is true, which we may or may not be able to formulate with precision, and we proceed to inquire whether it is true or not. For this purpose it is necessary to form a plan of investigation, and this is the most difficult part of the whole operation. We not only have to select the features of the diagram which it will be pertinent to pay attention to, but it is also of great importance to return again and again to certain features. (EP 2:212, 1903) 3. The word diagram is here used in the peculiar sense of a concrete, but possibly changing, mental image of such a thing as it represents. A drawing or model may be employed to aid the imagination; but the essential thing to be performed is the act of imagining. Mathematical diagrams are of two kinds; 1st, the geometrical, which are composed of lines (for even the image of a body having a curved surface without edges, what is mainly seen by the minds eye as it is turned about, is its generating lines, such as its varying outline); and 2nd, the algebraical, which are arrays of letters and other characters whose interrela-tions are represented partly by their arrangement and partly by repetitions. If these change, it is by instantaneous metamorphosis. (NEM 4:219) 4, We form in the imagination some sort of diagrammatic, that is, iconic, representation of the facts, as skeletonized as possible. The impression of the present writer is that with ordinary persons this is always a visual image, or mixed visual and muscular... This diagram, which has been constructed to represent intuitively or semi-intuitively the same relations which are abstractly expressed in the premisses, is then observed, and a hypothesis suggests itself that there is a certain relation between some of its parts -- or perhaps this hypothesis had already been suggested. In order to test this, various experiments are made upon the diagram, which is changed in various ways. (CP 2.778) 5. Diagrammatic reasoning is the only really fertile reasoning. If logicians would only embrace this method, we should no longer see attempts to base their science on the fragile foundations of metaphysics or a psychology not based on logical theory. (CP 4.571)
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